Central limit theorems for $U$-statistics of Poisson point processes

Abstract

A $U$-statistic of a Poisson point process is defined as the sum $\sum f(x_1,\ldots,x_k)$ over all (possibly infinitely many) $k$-tuples of distinct points of the point process. Using the Malliavin calculus, the Wiener-It\^{o} chaos expansion of such a functional is computed and used to derive a formula for the variance. Central limit theorems for $U$-statistics of Poisson point processes are shown, with explicit bounds for the Wasserstein distance to a Gaussian random variable. As applications, the intersection process of Poisson hyperplanes and the length of a random geometric graph are investigated.Comment: Published in at http://dx.doi.org/10.1214/12-AOP817 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

“…Univariate central limit theorems with bounds on the Wasserstein distance for Poisson functionals with finite Wiener-Itô chaos expansions and, in particular, Poisson Ustatistics are derived in [8], [9], and [16] using a general result from [14]. Our multivariate counterpart rests on a multivariate analogue in [13].…”

“…[16,Lemma 3.5]. The L 1 -version can be derived by approximation or by a direct calculation (just substitute (19) into (3) and observe that all resulting terms cancel out, except the integral representation (18) of F (i) t ).…”

“…(18). In fact, under some additional conditions, any Poisson functional with finite Wiener-Itô chaos expansion can be represented in such a way; see [16].…”

Gaussian limits of empirical multiparameter K-functions of homogeneous Poisson processes and tests for complete spatial randomness

“…So far the rates of convergence in the Theorem 4 as well as in Lemma 5 are unknown. It seems that the methods applied in the recent papers [36] and [30] can help to answer this open question. Proof of Theorem 2.…”

Stochastic Analysis for Poisson Point Processes

“…A U-statistic of order k with kernel f W X k ! R over a Poisson process is defined in [28] as X [7,13,15,28]). It turns out that the Wiener-Itô chaos expansion of a U-statistic is finite and thus Malliavin calculus is a particularly suitable method.…”